The secant-based detectable difference method was implemented using the accelerated greedy algorithm of M. Minoux [1]} to optimize the objective computed over all secants between points in the training data set consisting of \(N=750\)  snapshots of the shock-mixing layer velocity field.
We select among the 2210 sensor measurements of \(u\)  and \(v\)  velocity components on a grid of 1105 spatial locations taken directly from the shock-mixing layer snapshot data.
The target variables were chosen to be the leading two Isomap coordinates \({\mathbf {g}}({\mathbf {x}}) = \left( \phi _1({\mathbf {x}}), \phi _2({\mathbf {x}}) \right)\) , which reveal the phase angle \(\theta \) .
The greedy algorithm first reveals the two sensor locations marked by green stars and then the black star in Figure REF  over the range of \(0.02 \le \gamma \le 0.06\) , which can be used to reveal the exact phase of the system.
Choosing smaller values of \(\gamma \)  produce different sensors that can also be used to reveal the phase, but with reduced robustness to measurement perturbations.
Gaussian process regression [2]} was used to reconstruct the leading 100 principal components of the flowfields from the sensor measurements.
We used scikit learn's implementation which can be found at https://scikit-learn.org/stable/modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html together with a Matérn and white noise kernel whose parameters were optimized during the fit.
